Seminar on Fuchsian Groups (MRI Masterclass)
Practical Matters
This seminar is part of the MRI Masterclass on Arithmetic Geometry and Noncommutative Geometry. There is a maximal number of participants to the seminar of 13.
Location Utrecht, Minnaert Building room 021 (week 37+38), then Maths Building room 611AB (starting week 39)
Time Tuesdays 11-13; starts week 37 (7 Sep); ends week 51 (15 Dec)
Evaluation You are expected to give at least one seminar talk. You will study the material beforehand, hold a blackboard presentation about it, and maybe distribute lecture notes for your talk. You will pose one hand-in exercise to the other seminar participants, that they have to hand in at the next lecture. The hand-in exercise should be approved before by the seminar leaders. You are responsible for grading this hand-in exercise. You grade for the seminar is based on a grade that the seminar leaders give to your talk (40%), and on your homework grades (60%). Please note: giving a good seminar talk is extremely difficult; you should prepare your talks very well. For one thing, all future talks will depend on it. You should make sure that the material fits into the time slot, is presented in a clear way. You should make sure that you understand completely what you are talking about. If you can read some German, look at Manfred Lehn's advice here.
Prerequisites Undergraduate abstract algebra, geometry, topology and group theory.
Contents
Hyperbolic geometry, Fuchsian groups, Fundamental domains, Riemann surfaces, Co-compactness, first kind, Arithmetic Fuchsian groups; possibly: Kleinean groups and 3-dimensional geometry, Quantum chaos, modular surface.
Literature We will mainly follow the book [K] below; it is advisory to get a copy. We will also use some ad-hoc material from other sources.
Details
Week in yellow is current week. The schedule is still tentative (23 Jul).
| week | date | topic | References |
| 37 | 8 Sept | Introductory meeting & Dividing talks | - |
| 38 | 15 Sept | Hyperbolic Geometry I | [K] 1.1-1/3 |
| 39 | 22 Sept | Hyperbolic Geometry II | [K] 1.4-1.6 + ... |
| 40 | 29 Sept | Fuchsian Groups I | [K] 2.1-2.2 |
| 41 | 6 Okt | Fuchsian Groups II | [K] 2.3-2.4 |
| 42 | 13 Okt | Fundamental Regions I | [K] 3.1-3.3 |
| 43 | 20 Okt | Fundamental Regions II | [K] 3.4-3.5 (+3.6) |
| 44 | 27 Okt | Geometry of Fuchsian Groups I | [K] 4.1-4.2 |
| 45 | 27 Okt | No seminar | - |
| 46 | 10 Nov | Geometry of Fuchsian Groups II | [K] 4.3-4.4 |
| 47 | 17 Nov | Geometry of Fuchsian Groups III | [K] 4.5-4.6 |
| 48 | 24 Nov | Arithmetic Fuchsian Groups I | [K] 5.1-5.2 (+5.5) |
| 49 | 1 Dec | Arithmetic Fuchsian Groups II | [K] 5.3-5.4 (+5.6) |
| 50 | 8 Dec | Schottky groups: dynamics | [D] |
| 5 | 2 Feb | cancelled |
Main literature reference
[K] Svetlana Katok, Fuchsian Groups, Chicago Lectures in Mathematics, The University of Chicago Press, 1992, ISBN 0-226-42582-7
Other references
[B] Lipman Bers, Finitely generated Kleinian groups. An introduction. Ann. Acad. Sci. Fenn. Ser. A I Math. 13 (1988), no. 3, 313--327 (article).
[D] Francoise Dal'bo, Trajectoires geodesiques et horocycliques, EDP Science/CNRS Editions, 2007, ISBN 978-2-86883-997-8
[M] Bernard Maskit, Kleinian groups. Grundlehren der Mathematischen Wissenschaften, vol. 287, Springer-Verlag, Berlin, Heidelberg, New York, 1988, ISBN 3-540-178746-9.
[MSW] David Mumford, Caroline Series, David Wright,
Indra's pearls.
The vision of Felix Klein. Cambridge University Press, New York, 2002, ISBN: 0-521-35253-3.
[T] William P. Thurston, Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997, ISBN: 0-691-08304-5.
Additional references
[FK] H. Farkas, I. Kra, Riemann Surfaces, Graduate Texts in Math 71,
Springer Verlag
[G] R.C. Gunning, Lectures on Modular Forms,
Ann of Math Studies 48, Princeton University Press
[Le]
J. Lehner, Discontinuous Groups and Automorphic Functions, Mathematical
Surveys and Monographs 8, Am Math Soc (part of
it here on books.google.com)
[M2] J. Milne, Modular Functions and Modular Forms
(available from www.jmilne.org)
[Sc] M. Schlichenmaier, Introduction to Riemann Surfaces,
Algebraic Curves and Moduli Spaces, Theoretical and Math Phys, Springer
Verlag
[S] G. Shimura, Introduction to the arithmetic
theory of automorphic functions, Princeton University Press
[Y] M. Yoshida, Hypergeometric Functions, My Love, Aspects of Math E32,
Vieweg Verlag, Wiesbaden